Engineering Applications of General Relativity

Suppose the metric is planar, ie, of the form d τ 2 = g 00 ( r , ϕ ) d t 2 + g 11 ( r , ϕ ) d r 2 + g 22 ( r , ϕ ) d ϕ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0329.tif"/> Then, motion takes place in the r – ϕ plane. The Lagrangian for the motion of light which follows the null geodesic equations is given by L ( r , ϕ , t ′ , r ′ , ϕ ′ ) = τ ′ = ( g 00 t ′ 2 + g 11 r ′ 2 + g 22 ϕ ′ 2 ) 1 / 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0330.tif"/> where ξ = dξ/dλ with λ a parameter that varies along the geodesic curve. A priori, we do not set τ ′ = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/ieq0049.tif"/> for the null geodesic condition. Only after all the equations of motion are setup do we impose this condition. The reason for this is that the integrals of the equations of motion are expressed in terms of constants that become infinite when the null geodesic condition is imposed and yet some of the ratios of these infinite constants is finite. To see this explicitly, we write down the Euler-Lagrange equations as d d λ L , t ′ = L , t = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0331.tif"/> so that L , t ′ = K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0332.tif"/> where K is a constant, ie, g 00 d t / d τ = K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0333.tif"/> 46Likewise, d d λ L , ϕ ′ = L , ϕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0334.tif"/> ( d / d λ ) ( g 22 d ϕ / d τ ) = L , ϕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0335.tif"/> and d d λ L , r ′ = L , r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0336.tif"/> gives ( d / d λ ) ( g 11 d r / d τ ) = L , r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0337.tif"/> The ϕ equation can be expressed as ϕ ′ ( d / d ϕ ) ( g 22 ϕ , τ ) = L , ϕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0338.tif"/> Consider now the particular case when the metric coefficients do not depend on ϕ. Then, L_,ϕ = 0 and we get g 22 ϕ , τ = β https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0339.tif"/> In the null geodesic limit when dτ = 0, both K and β become infinite constants but since ( g 22 / g 00 ) ( d ϕ / d t ) = K / β https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0340.tif"/> is finite, we can write ( g 22 ( r ) / g 00 ( r ) ) d ϕ / d t = A − − − ( 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0341.tif"/> where A is a finite constant and the trajectory equation is then given by (1) and 0 = ( d τ / d t ) 2 = g 00 ( r ) + g 11 ( r ) ( d r / d t ) 2 + g 22 ( r ) ( d ϕ / d t ) 2 − − − ( 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0342.tif"/> Thus the trajectory equation is 0 = g 00 ( r ) + g 11 ( r ) ( d r / d ϕ ) 2 A 2 ( g 00 ( r ) / g 22 ( r ) ) 2 + g 22 ( r ) A 2 ( g 00 ( r ) / g 22 ( r ) ) 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0343.tif"/> This equation can be integrated to give r as a function of ϕ. When we integrate this equation, we obtain r as a function of ϕ, A and another constant, say B of integration. So we can write r = r ( ϕ , A , B ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0344.tif"/> Now, the transmitter transmits a photon sitting at the point (r ₀,ϕ ₀) and he keeps changing the angle of the photon gun until he gets a signal of reception from the receiver located at (r ₁,ϕ ₁). This signal is received, when the transmitter orientation defined by dr/dϕ|r ₀, _{ϕ 0} = a is measured by the transmitter. The transmitter also conveys the coordinate time t ₀ at which he sends the photon and likewise, the receiver conveys to the transmitter the coordinate time t ₁ at which he receives the photon. Now the equations of motion gives d ϕ ( t ) / d t = A g 00 ( r ( t ) ) / g 22 ( r ( t ) ) = A g 00 ( r ( ϕ ( t ) , A , B ) ) / g 22 ( r ( ϕ ( t ) , A , B ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0345.tif"/> 47which is integrated to give ϕ ( t ) = ϕ ( t , A , B , C ) , r ( t ) = r ( ϕ ( t ) , A , B ) = r ( t , A , B , C ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0346.tif"/> where C is another constant of integration. Now, the constants A,B,C are determined from ϕ 0 = ϕ ( t 0 ) = ϕ ( t 0 , A , B , C ) , r 0 = r ( t 0 ) = r ( t 0 , A , B , C ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0347.tif"/> ϕ 1 = ϕ ( t 1 ) = ϕ ( t 1 , A , B , C ) , r 1 = r ( t 1 ) = r ( t 1 , A , B , C ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0348.tif"/> These are four equations and should be combined with the fifth equation d r ( ϕ 0 , A , B ) / d ϕ = a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003173021/977befbb-7737-4789-b58c-3e5dd47d8927/content/eqn0349.tif"/> to solve for the five variables A,B,C,r ₁,ϕ ₁. In particular, the range and bearing r ₁,ϕ ₁ of the receiver are obtained by the transmitter.